Ellipsoidal Distributions of Charge or Mass

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Abstract
The Coulomb (or gravitational) energy is calculated for a distribution of charge (or mass) in which the surfaces of constant density are a family of similar concentric ellipsoids. The density can vary in any manner from one surface to another, and the ellipsoids need not have an axis of symmetry. Two examples are discussed: the charge distribution of a deformed atomic nucleus having a diffuse surface, and the mass distribution of a stellar galaxy. The energy is shown to be a product of two factors. One is the energy of the spherical distribution from which the ellipsoidal distribution can be obtained by a volume‐preserving deformation. The other is an anisotropy factor that has a simple geometrical significance and depends only on the two eccentricities of the ellipsoids. Its values range from unity to zero and are tabulated numerically.